Splitting, degenerate states

Figure 1- Representatiori of degenerate states from nonrelativistic components, (a) Degenerate zeroth-order states at (b) Spin-orbit interaction splits 11 state, (c) With full spin-orbit...

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

For fixed va,vb,vc and 7, there are 27 + 1 initially degenerate states with /IT = 0, 1, 2,..., 7. The rotation-vibration interaction splits these states. The matrices to diagonalize are of the type ... 

The splitting of the free-ion term in octahedral symmetry Oh symmetry) reduces the degeneracy of the five d orbitals. Three orbitals have energy lower than the other two. This means that if the orbitals are populated by one electron, three degenerate states are possible, according to the three possible positions for the electron in the low-energy levels (T symmetry) ... 

The perturbation of a free-ion term, specified by L and S, by spin—orbit coupling takes the form of a splitting into states, 2L + 1 or 25 + 1 in number, whichever is the less. Each state is specified by the total angular momentum quantum number J, and is (2J + l)-fold degenerate. J takes on values from L—5 to L+5, and for a shell less than half-full the state with the lowest value of J lies lowest in energy. The energy separation between a state specified by J and that lying next above it, specified by J + 1, is... 

Only in spatially degenerate states, M so may cause a multiplet splitting in first order. 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

A number of interesting facts emerge from this treatment. It is seen that P states remain threefold degenerate in cubic fields, but are split by a tetragonal distortion into a twofold and singlefold degenerate state. For a D state, or one d electron outside of closed... 

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